Why vertices are changing their position during Weight Paint Mode?
Via weight painting, you increase or decrease influence of the bones on vertices. That's why they change position. If f.ex. some bone influences a vertice with strength of 0 nothing happens, but when you give this vertice weight of 1 it starts to be influenced by this bone with strength of 1 and it's moving closer to it. This is normal and expected behaviour. Although your example looks weird.Could you provide the .blend?